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Lie Groups Fall 2016 (Cohen) Lecture Notes

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LIE GROUPS FALL 2016 (COHEN) LECTURE NOTES 1. Preliminaries Remark 1.1. The content in these notes is Frankensteined together from many sources, including Knapp's Lie Groups Beyond An Introduction, Bump's Lie Groups, Tao's Hilbert's Fifth Problem and Related Topics, Varadarajan's Lie Groups, Lie Algebras, and Their Representations, Hilgert and Neeb's Structure and Geometry of Lie Groups, and the author's own paltry brain. Although I have added details to many proofs or changed them entirely, I am not making any serious attempt to avoid plagiarism and primary credit belongs to those authors. In particular many of the exercises appear in Knapp. 1.1. Topological Groups. Definition 1.2. A topological group is a pair (G; τ) where G is a group, and τ is a topology on G satisfying the following conditions: (1) the mapping (g; h) 7! gh, G × G ! G is continuous (with respect to τ); and (2) the mapping g 7! g−1, G ! G is continuous (with respect to τ). We will always refer to the identity element of G as eG, or, if the context is clear, just as e. Remark 1.3. Most of the time the particular topology τ will be assumed or understood in context, and we will just refer to G itself as a topological group (rather than the pair (G; τ)). Example 1.4 (Examples of Topological Groups). Any abstract group equipped with the discrete topol- ogy; any abstract group equipped with the trivial topology; (Z; +); (Q; +); (R; +); (Rn; +); (R+; ·); the circle group S1; GL(n; R); GL(n; C); SL(n; R); SL(n; C); any Banach space regarded as an additive group equipped with the norm topology; many homeomorphism groups equipped with the compact-open topology; .... Exercise 1. Let G be a group and τ a topology on G. Prove that (G; τ) is a topological group iff the mapping (g; h) 7! gh−1, G × G ! G is continuous with respect to τ. Exercise 2. Let G be a topological group, and let h 2 G be arbitrary. Define the mappings i : G ! G, i(g) = g−1; Lh : G ! G, Lh(g) = hg; Rh : G ! g, Rh(g) = gh. Prove that i, Lh, and Rh are self-homeomorphisms of G. Definition 1.5. Let G be a group and let A; B ⊆ G. Let n be a positive integer. We define A−1 = fa−1 : a 2 Ag; AB = fab : a 2 A; b 2 Bg; n A = fa1a2:::an : a1; a2; :::; an 2 Ag. A is called symmetric if A = A−1. Exercise 3. Let G be a topological group and let U ⊆ G be an open neighborhood of e. Prove that there exists an open neighborhood V of e such that V 2 ⊆ U. Exercise 4. Let G be a topological group and let U ⊆ G be an open neighborhood of e. Prove that for every positive integer n, there exists an open neighborhood V of e such that V n ⊆ U. Exercise 5. Let G be a topological group and let U ⊆ G be an open neighborhood of e. Prove that for every positive integer n, there exists a symmetric open neighborhood V of e such that V n ⊆ U. 1 2 LIE GROUPS FALL 2016 (COHEN) LECTURE NOTES 1.2. Differentiable Manifolds. Definition 1.6. A smooth premanifold of dimension n (n 2 N) is a Hausdorff topological space M together with a set U of pairs (U; φ) (called charts), where U is an open subset of M and φ : U ! Rn is a mapping, satisfying (i) the collection fU : 9φ(U; φ) 2 Ug is an open cover of M; (ii) for each (U; φ) 2 U, φ is a homeomorphism of U onto an open subset of Rn; and (iii) for each (U; φ); (V; ) 2 U the map ◦φ−1 is a smooth diffeomorphism from φ(U \V ) onto (U \V ). The set U is called a preatlas. If M and N are smooth premanifolds, a continuous map f : M ! N is called smooth if when- ever (U; φ) and (V; ) are charts of M and N respectively, the map ◦ f ◦ φ−1 is a smooth map from φ(U \ f −1(V )) into (V ). The smooth map f is called a diffeomorphism if it is a bijection and has a smooth inverse. If M is a smooth premanifold with preatlas U, then we let U 0 denote the set of all pairs (U; φ) where U is an open subset of M and φ is a diffeomorphism of U onto an open subset of Rn. If U = U 0, then we call U an atlas and M a smooth manifold. Suppose M is a smooth manifold and m 2 M. Suppose U is an open neighborhood of m and (U; φ) 2 U, where φ(m) is the origin in Rn. Then we may write φ(u) = (x1(u); x2(u); :::; xn(u)) for u 2 U where the xi's (1 ≤ i ≤ n) are smooth functions from U to R. In this case we call the tuple (x1; ::; xn) local coordinates at m or coordinate functions on U. The set U is called a coordinate patch or coordinate neighborhood. Exercise 6. Show that if M is a smooth manifold and m 2 M, then there exist local coordinates at m. Exercise 7. Show that if f : N ! P and g : M ! N are smooth maps between smooth manifolds M; N; P , then the composition f ◦ g is also a smooth map from M into P . Fact 1.7. If M and N are smooth manifolds, then the product M × N can be made into a smooth premanifold in a natural way by defining the charts in M × N to be products of the charts in M and N, respectively. M ×N then admits a natural smooth manifold structure, in such a way that m 7! (m; n) is a smooth diffeomorphism of M onto M ×fng for each n 2 N, and n 7! (m; n) is a smooth diffeomorphism of N onto fmg × N. Definition 1.8. A (real) Lie group is a separable topological group G which is also a smooth manifold, satisfying the following conditions: (1) the mapping (g; h) 7! gh, G × G ! G is smooth; and (2) the mapping g 7! g−1, G ! G is smooth. Proposition 1.9. Let G be a Lie group. The maps i, Lh, and Rh (h 2 G) defined in Exercise 2 are self-diffeomorphisms of G. Proof. We already know i, Lh, and Rh are self-homeomorphisms, so we need only show they are each smooth with smooth inverse. Since i is its own inverse, it has a smooth inverse. For any h 2 G, for every g 2 G, we have Lh(g) = hg = ·(h; g), so Lh is the composition of the smooth map g 7! (h; g) with smooth multiplication. Therefore Lh is smooth, and similarly for Rh. The inverses of Lh and Rh are Lh−1 and Rh−1 , respectively, which are smooth maps by the previous two sentences. 2. Classical Lie Groups and Lie Algebras 2.1. Linear Groups. Definition 2.1. The general linear groups are the groups GL(n; R) of nonsingular n×n real matrices, and GL(n; C) of nonsingular n × n complex matrices, for n 2 Z+. The underlying group operation for each of these groups is matrix multiplication. We also define: LIE GROUPS FALL 2016 (COHEN) LECTURE NOTES 3 SL(n; C) = fX 2 GL(n; C) : det X = 1g SL(n; R) = fX 2 GL(n; R) : det X = 1g t O(n) = fX 2 GL(n; R): XX = 1g t SO(n) = fX 2 GL(n; R): XX = 1; det X = 1g ∗ U(n) = fX 2 GL(n; C): XX = 1g ∗ SU(n) = fX 2 GL(n; C): XX = 1; det X = 1g The above, respectively, are the special linear groups over C and over R, the orthogonal groups, the special orthogonal groups, the unitary groups, and the special unitary groups for n 2 Z+. We will always denote the identity matrix by I, or In if the context is not clear. Remark 2.2. For each n 2 Z+, the general linear group GL(n; C) is an open subset of the Euclidean n2 2n2 space Matn(C) of n×n complex matrices (which we identify with C or R ). So GL(n; C) is a smooth manifold in a natural way, and a Lie group under this smooth structure. Similarly, each of the other groups listed in the previous definition sits inside GL(n; C) as a closed subgroup, and becomes a smooth manifold by inheriting the structure of GL(n; C). Each group then can be viewed as a Lie group. 2.2. The Lie Algebra of a Linear Group. Definition 2.3. Let G be a closed subgroup of GL(n; C). A smooth curve in G is a smooth mapping c : J ! G, where J is a subinterval of R with nonempty interior. The Lie algebra of G is the set g = fc0(0) : c is a smooth curve in G with c(0) = Ig. So g is a set of matrices of the same size as G. The members of g are not necessarily invertible. Example 2.4. If G = SO(3), then 2 cos t sin t 0 3 c(t) = 4 − sin t cos t 0 5 for t 2 (−1; 1) 0 0 1 is an example of a smooth curve in G with c(0) = I3.
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